Sylvia Byrd

2022-07-02

How can the energy of a magnetic field spread at speed of light?
I would like to solve one question. Suppose that we have a point charge. Then we apply a force on it, and as it accelerates, it emits electromagnetic waves and is pushed back due to Abraham-Lorentz force. But then, when we drop it and it moves at constant speed, it starts to create a magnetic field that spreads at speed of light. So as the field occupies more volume, it increases its energy but we are not giving any energy as we have dropped it. So, where does this energy come from?

Nicolas Calhoun

As you've probably noticed, the formula
$Energy\sim \int {c}^{2}|\stackrel{\to }{B}{|}^{2}+|\stackrel{\to }{E}{|}^{2}dV$
appears to change with time.
But it's a bit more complicated than that. The electric field of a moving particle is different from that of a stationary particle (going as ${\stackrel{\to }{E}}_{\perp }^{\prime }=\mathrm{cosh}\varphi \cdot {\stackrel{\to }{E}}_{\perp }$ after lorentz transform). So while the magnetic field is getting "refreshed" and increasing, the electric field is getting refreshed and overall decreasing in such a way that the total energy remains constant.
That doesn't mean, however, that the total energy stored in the fields remains the same, ${E}^{2}-{c}^{2}{B}^{2}$ is the only lorentz invariant quality that will be the same for a stationary vs. moving point charge.

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